Quesiton 1: John's annual income (in dollars) follows a two-pointmixture distribution, where 30% has a Pareto distribution with mean 1,000 and variance 3,000,000, and 70% has an Exponential distribution with mean 60,000.
(a) Calculate John's expected annual income.
(b) Calculate the variance of his annual income.
(c) Calculate the coef?cient of skewness of his annual income.
Quesiton 2: The number of accidents in a highway during a month follows a Poisson distribution with mean , where follows an Exponential distribution with mean 5. Calculate the probability of only having one accident this month.
Quesiton 3: The future lifetime of a random individual (in years) has constant force of mortality , where has a Uniform distribution on the interval [0:025; 0:04].
(a) Calculate the probability that a random individual does not survive 30 years from today.
(b) Using the normal approximation, calculate the probability that a random individual does not survive 30 years. Compare to above.
(c) An individual buys a life insurance policy for 1 million dollars, and keeps it until her death. Calculate the actuarial present value of the policy's death bene?t. Consider the annual continuously compounded interest rate to be 3%.